Discrete versions of the Beckman-Quarles theorem from the definability results of Raphael M. Robinson
We derive author's discrete forms of the Beckman-Quarles theorem from the definability results of Raphael M. Robinson.
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arXiv subjects
Apoloniusz Tyszka
Publications and source records attributed to Apoloniusz Tyszka.
A discrete form of the Beckman-Quarles theorem for two-dimensional strictly convex normed spaces
Let X be a real normed vector space and dim X \ge 2. Let d>0 be a fixed real number. We prove that if x,y \in X and ||x-y||/d is a rational number then there exists a finite set {x,y} \subseteq S(x,y) \subseteq X with the following property: for each strictly convex Y of dimension 2 each map from S(x,y) to Y preserving the distance d preserves the distance between x and y. It implies that each map from X to Y that preserves the distance d is an isometry.
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A discrete form of the Beckman-Quarles theorem for rational eight-space
Let Q denote the field of rational numbers. Let F \subseteq R is a euclidean field. We prove that: (1) if x,y \in F^n (n>1) and |x-y| is constructible by means of ruler and compass then there exists a finite set S(x,y) \subseteq F^n containing x and y such that each map from S(x,y) to R^n preserving unit distance preserves the distance between x and y, (2) if x,y \in Q^8 then there exists a finite set S(x,y) \subseteq Q^8 containing x and y such that each map from S(x,y) to R^8 preserving unit distance preserves the distance between x and y.
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On binary relations without non-identical endomorphisms
On every set A there is a rigid binary relation, i.e. such a relation R that there is no homomorphism (A,R)->(A,R) except the identity (Vopenka et al. [1965]). We state two conjectures which strengthen this theorem. We prove these conjectures for some cardinalities.
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A combinatorial characterization of second category subsets of X^ω
Let a finite non-empty X is equipped with discrete topology. We prove that S \subseteq X^ωis of second category if and only if for each f:ω-> \bigcup_{n \in ω} X^n there exists a sequence {a_n}_{n \in ω} belonging to S such that for infinitely many i \in ωthe infinite sequence {a_{i+n}}_{n \in ω} extends the finite sequence f(i).
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A new combinatorial characterization of the minimal cardinality of a subset of R which is not of first category
Let M denote the ideal of first category subsets of R. We prove that min{card X: X \subseteq R, X \not\in M} is the smallest cardinality of a family S \subseteq {0,1}^ωwith the property that for each f: ω-> \bigcup_{n \in ω}{0,1}^n there exists a sequence {a_n}_{n \in ω} belonging to S such that for infinitely many i \in ωthe infinite sequence {a_{i+n}}_{n \in ω} extends the finite sequence f(i). We inform that S \subseteq {0,1}^ωis not of first category if and only if for each f: ω-> \bigcup_{n \in ω}{0,1}^n there exists a sequence {a_n}_{n \in ω} belonging to S such that for infinitely many i \in ωthe infinite sequence {a_{i+n}}_{n \in ω} extends the finite sequence f(i).
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On the minimal cardinality of a subset of R which is not of first category
Let M be the ideal of first category subsets of R and non(M)=min{card X: X \subseteq R, X \not\in M}. We consider families Φof sequences converging to \infty, with the property that for every open set U \subseteq R that is unbounded above there exists a sequence belonging to Φ, which has an infinite number of terms belonging to U. We present assumptions about Φwhich imply that the minimal cardinality of Φequals non(M).
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Discrete versions of the Beckman-Quarles theorem
TO APPEAR IN AEQUATIONES MATHEMATICAE - WITHOUT THEOREM 2. THEOREM 2 IS CORRECTLY PROVED IN PREVIOUS VERSIONS 1 AND 2. AUTHOR'S VERSION 3 (WITH A NEW FIGURE 6A) IS UNNECESSARY. Let F \subseteq R denote the field of numbers which are constructible by means of ruler and compass. We prove that: (1) if x,y \in R^n (n>1) and |x-y| is an algebraic number then there exists a finite set S(x,y) \subseteq R^n containing x and y such that each map from S(x,y) to R^n preserving all unit distances preserves the distance between x and y; if x,y \in F^n then we can choose S(x,y) \subseteq F^n, (2) only algebraic distances |x-y| have the property from item (1), (3) if X1,X2,...,Xm \in R^n (n>1) lie on some affine hyperplane then there exists a finite set L(X1,X2,...,Xm) \subseteq R^n containing X1,X2,...,Xm such that each map from L(X1,X2,...,Xm) to R^n preserving all unit distances preserves the property that X1,X2,...,Xm lie on some affine hyperplane, (4) if J,K,L,M \in R^n (n>1) and |JK|=|LM| (|JK|<|LM|) then there exists a finite set C(J,K,L,M) \subseteq R^n containing J,K,L,M such that any map f:C(J,K,L,M) \to R^n that preserves unit distance satisfies |f(J)f(K)|=|f(L)f(M)| (|f(J)f(K)|<|f(L)f(M)|).
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On a combinatorial property of families of sequences converging to +infinity
We consider families F of sequences converging to +infinity that F satisfies the following condition (C): (C): if an open set U in the real line is unbounded above then there exists a sequence belonging to F, which has an infinite number of terms belonging to U. For the functions f,g from {0,1,2,...} to {0,1,2,...} we define: f =< g if and only if {i: f(i) > g(i)} is finite. Let b denote the smallest cardinality of a unbounded (in the sense of =<) family of functions from {0,1,2,...} to {0,1,2,...}, see [1]. Theorem 1. If F is a family of sequences converging to +infinity and card F < b, then F does not satisfy condition (C). Corollary. Every family of sequences converging to +infinity which satisfies the condition (C) is uncountable, Martin's axiom implies that each such family has cardinality continuum (because Martin's axiom implies that b=continuum, see [1], [2]). Theorem 2. There exists a family of sequences converging to +infinity which satisfies condition (C) and has cardinality b. References [ 1 ] R. Frankiewicz, P. Zbierski, Hausdorff gaps and limits, Amsterdam: North-Holland, 1994. [ 2 ] T. Jech, Set theory, New York, Academic Press, 1978.
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